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Local geometry of high-dimensional mixture models: Effective spectral theory and dynamical transitions

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arXiv 2025年

作者： Arous, Gerard Ben Gheissari, Reza Huang, Jiaoyang Jagannath, Aukosh Courant Institute New York University United States Department of Mathematics Northwestern University United States Department of Statistics and Data Science University of Pennsylvania United States Department of Statistics and Actuarial Science Department of Applied Mathematics Cheriton School of Computer Science University of Waterloo Canada

We study the local geometry of empirical risks in high dimensions via the spectral theory of their Hessian and information matrices. We focus on settings where the data, (Y)n=1 ∈ d, are i.i.d. draws of a k-component Gaussian mixture model, and the loss depends on the projection of the data into a fixed number of vectors, namely xY, where x ∈ d×C are the parameters, and C need not equal k. This setting captures a broad class of problems such as classification by one and two-layer networks and regression on multi-index models. We prove exact formulas for the limits of the empirical spectral distribution and outlier eigenvalues and eigenvectors of such matrices in the proportional asymptotics limit, where the number of samples and dimension n, d → ∞ and n/d = ∈ (0, ∞). These limits depend on the parameters x only through the summary statistic of the (C + k) × (C + k) Gram matrix of the parameters and class means, G = (x, µ)(x, µ). It is known that under general conditions, when x is trained by stochastic gradient descent, the evolution of these same summary statistics along training converges to the solution of an autonomous system of ODEs, called the effective dynamics. This enables us to connect the spectral theory to the training dynamics. We demonstrate our general results by analyzing the effective spectrum along the effective dynamics in the case of multi-class logistic regression. In this setting, the empirical Hessian and information matrices have substantially different spectra, each with their own static and even dynamical spectral transitions. Copyright © 2025, The Authors. All rights reserved.

关键词： Stochastic systems

Lung Cancer Diagnosis Based on Object Detection Technology 14th

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Lung Cancer Diagnosis Based on Object Detection Technology

14th International Conference on Frontier Computing, FC 2024

作者： Xu, Jia-Lang Wang, Pei-Yun Huang, Tzu-Hsuan Hsu, Ying-Lin Huang, Kuo-Yang Chen, Mu-Yen Artificial Intelligence Development Center Changhua Christian Hospital Changhua Taiwan Doctoral Program in Data Science and Industrial Analytics National Chung Hsing University Taichung Taiwan Department of Applied Mathematics and Institute of Statistics National Chung Hsing University Taichung Taiwan Division of Chest Medicine Department of Internal Medicine Changhua Christian Hospital Changhua Taiwan Department of Engineering Science National Cheng Kung University Tainan Taiwan

ISBN: (纸本)9789819623945

With the rise of artificial intelligence, many people nowadays use artificial intelligence to help solve some problems in life, and the medical field is also with the rise of artificial intelligence, many people are slowly applying artificial intelligence to medical images to assist in decision-making, this study proposes a Yolov8 algorithm for object tracking of CT images of lung cancer, which can quickly tell the doctor where the cancer is and what the type of cancer is. The results of this study show that the Box_Loss, Cls_Loss, and Dfl_loss of the Yolov8 algorithm in detecting the location of lung cancer are 1.3689, 0.4552, and 1.5569 in the validation data, respectively, and thus this study suggests that this method can help physicians know the location of the tumor and what kind of tumor it belongs to. Therefore, this study suggests that this method can help physicians know the location of the tumor and the type of tumor. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.

关键词： Lung cancer

Unbiased Parameter Estimation for Bayesian Inverse Problems

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arXiv 2025年

作者： Chada, Neil K. Jasra, Ajay Maama, Mohamed Tempone, Raul Department of Mathematics City University of Hong Kong China School of Data Science The Chinese University of Hong Kong Shenzhen CN Shenzhen China Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximizer of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

Natural averaging may complement known biological constraints in sexual reproduction’s advantages over asexual in conserving species quantitative traits

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Scientific Reports 2025年第1期15卷 1-13页

作者： Marron, Assaf Szekely, Smadar Cohen, Irun R. Harel, David Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100 Israel Department of Immunology and Regenerative Biology Weizmann Institute of Science Rehovot 76100 Israel

Commonly recognized effects of sexual reproduction include increased diversity, improved adaptability, enabling of DNA repair, constrained accumulation of deleterious mutations, and species genotype homogenization. Additionally, there are studies that show how sexual reproduction slows down certain evolutionary responses, offering advantages in population cumulative growth and stability over time and other metrics. Here, we contribute an observation of another distinct effect of sexual reproduction, focusing on retaining a species’s key traits. In an initial mathematical analysis and simulation, we show that in an environment where copying is prone to error, quantitative polygenic traits that are shared within a parents’ generation are transmitted to future generations under sexual reproduction with less deviation than under asexual reproduction. Furthermore, the model shows that this retention of common traits (abbr. RoCT), is driven by the very nature of mixing of parental traits, and occurs even before adding effects like trait-specific reproductive advantages, DNA repair, or the raising of reproductive barriers. Since survival of ecosystems depends on the ability of individuals to replace the networked interactions and interdependencies associated with failing, dying, or absent members of the same species, RoCT helps sustain species and ecosystems. © The Author(s) 2025.

关键词： Asexual reproduction Bi-parental Clonal reproduction Probabilistic sampling Sexual reproduction

Teaching An Old Dog New Tricks: Porting Legacy Code to Heterogeneous Compute Architectures With Automated Code Translation

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arXiv 2025年

作者： Nytko, Nicolas Reisner, Andrew Moulton, J. David Olson, Luke N. West, Matthew Department of Computer Science University of Illinois at Urbana-Champaign United States Applied Computer Science Los Alamos National Laboratory New Zealand Applied Mathematics and Plasma Physics Los Alamos National Laboratory New Zealand Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign United States

Legacy codes are in ubiquitous use in scientific simulations;they are well-tested and there is significant time investment in their use. However, one challenge is the adoption of new, sometimes incompatible computing paradigms, such as GPU hardware. In this paper, we explore using automated code translation to enable execution of legacy multigrid solver code on GPUs without significant time investment and while avoiding intrusive changes to the codebase. We developed a thin, reusable translation layer that parses Fortran 2003 at compile time, interfacing with the existing library Loopy (Klöckner 2014) to transpile to C++/GPU code, which is then managed by a custom MPI runtime system that we created. With this low-effort approach, we are able to achieve a payoff of an approximately 2–3× speedup over a full CPU socket, and 6× in multi-node settings. © 2025, CC BY-SA.

关键词： Investments

A New Dominating Set Game on Graphs

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arXiv 2025年

作者： Fiscus, Sean Hurlbert, Glenn Myzelev, Eric Pence, Travis Department of Mathematics Duke University DurhamNC United States Department of Mathematics and Applied Mathematics Virginia Commonwealth University RichmondVA United States Department of Mathematics University of Pennsylvania PhiladelphiaPA United States Department of Computer Science University of Wisconsin MadisonWI United States

We introduce a new two-player game on graphs, in which players alternate choosing vertices until the set of chosen vertices forms a dominating set. The last player to choose a vertex is the winner. The game fits into the scheme of several other known games on graphs. We characterize the paths and cycles for which the first player has the winning strategy. We also create tools for combining graphs in various ways (via graph powers, Cartesian products, graph joins, and other methods) for building a variety of graphs whose games are won by the second player, including cubes, multidimensional grids with an odd number of vertices, most multidimensional toroidal grids, various trees such as specialized caterpillars, the Petersen graph, and others. Finally, we extend the game to groups and show that the second player wins the game on abelian groups of even order with canonical generating set, among *** Codes 91A43, 05C57 Copyright © 2025, The Authors. All rights reserved.

关键词： Graphic methods

Integration of aggregated data in causally interpretable meta-analysis by inverse weighting

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arXiv 2025年

作者： Vo, Tat Thang Le, Tran Trong Khoi Afach, Sivem Vansteelandt, Stijn Research Group EPIDERME Faculty of Medicine University Paris Est Creteil France Department of Applied Mathematics Computer Science and Statistics Ghent University Belgium

Obtaining causally interpretable meta-analysis results is challenging when there are differences in the distribution of effect modifiers between eligible trials. To overcome this, recent work on transportability methods has considered standardizing results of individual studies over the case-mix of a target population, prior to pooling them as in a classical random-effect meta-analysis. One practical challenge, however, is that case-mix standardization often requires individual participant data (IPD) on outcome, treatments and case-mix characteristics to be fully accessible in every eligible study, along with IPD case-mix characteristics for a random sample from the target population. In this paper, we aim to develop novel strategies to integrate aggregated-level data from eligible trials with non-accessible IPD into a causal meta-analysis, by extending moment-based methods frequently used for population-adjusted indirect comparison in health technology assessment. Since valid inference for these moment-based methods by M-estimation theory requires additional aggregated data that are often unavailable in practice, computational methods to address this concern are also developed. We assess the finite-sample performance of the proposed approaches by simulated data, and then apply these on real-world clinical data to investigate the effectiveness of risankizumab versus ustekinumab among patients with moderate to severe psoriasis. © 2025, CC BY.

关键词： Inverse problems

Fractional order modeling of dengue transmission dynamics in Bangladesh

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Partial Differential Equations in applied mathematics 2025年 14卷

作者： Sikder, Arun Kumar Islam, Md Hamidul Department of Computer Science and Engineering Varendra University Rajshahi 6204 Bangladesh Department of Applied Mathematics University of Rajshahi Rajshahi 6205 Bangladesh

Over the past three decades, the epidemiological characteristics of dengue fever in Bangladesh have raised significant public health concerns, with the disease progressively spreading across the country's subtropical and tropical regions since 2004. This study introduces a fractional-order mathematical model that incorporates both human and mosquito populations using Caputo derivatives to account for memory and hereditary effects in disease transmission. Analytical solutions are approximated using the Laplace–Adomian decomposition method, while the Adams–Bashforth-Moulton predictor–corrector (PECE) scheme is applied for numerical solutions. Model parameters are estimated via the maximum likelihood method, using dengue case data from Bangladesh (June 1, 2022–September 30, 2022). Our findings indicate that the fractional-order model effectively captures a range of transmission scenarios, with lower derivative orders correlating with reduced outbreak severity. Moreover, numerical solutions exhibit higher accuracy compared to analytical approximations, particularly for complex nonlinear systems. Additionally, the results suggest that controlling mosquito populations, reducing transmission rates, and improving treatment measures are critical strategies for dengue mitigation. © 2025 The Authors

关键词： Dengue fever Fractional order modeling LADM Mathematical modeling MLE fitting Monte Carlo method PECE method

Enhancing Thyroid Disease Prediction Using Machine Learning: A Comparative Study of Ensemble Models and Class Balancing Techniques

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Research Square

Research Square 2025年

作者： Zhong, Jiachen Wang, Yiting Department of Applied Mathematics University of Washington SeattleWA98105 United States Department of Computer Science Florida International University MiamiMI33199 United States

Thyroid disease is a significant medical condition affecting approximately 20 million Americans. The thyroid gland regulates metabolism through hormones such as triiodothyronine (T3) and thyroxine (T4), with disorders typically manifesting as hypothyroidism or hyperthyroidism. This study evaluates the performance of various machine learning models in predicting and diagnosing thyroid disease, including Logistic Regression, Decision Trees, Random Forest, XGBoost, Support Vector Machines, Neural Networks, Bagging, and Stacking methods. The bagging model utilizing three decision trees achieved the highest F1 Score of 0.9766, outperforming both Random Forest and XGBoost. Additionally, experiments on class balancing through undersampling and regrouping significantly improved model performance, particularly for stacking models with XGBoost, which attained an F1 Score of 0.9944. © 2025, CC BY.

关键词： Contrastive Learning